 # Journal of Nonlinear Mathematical Physics

Volume 27, Issue 4, September 2020, Pages 633 - 646

# Finite genus solutions to the lattice Schwarzian Korteweg-de Vries equation

Authors
Xiaoxue Xu*
School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, 450001, People’s Republic of China, xiaoxuexu@zzu.edu.cn
Cewen Cao
School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, 450001, People’s Republic of China, cwcao@zzu.edu.cn
Guangyao Zhang
School of Science, Huzhou University, Zhejiang, 313000, People’s Republic of China, zgy101003@163.com
*Corresponding author.
Corresponding Author
Xiaoxue Xu
Received 22 March 2019, Accepted 25 January 2020, Available Online 4 September 2020.
DOI
https://doi.org/10.1080/14029251.2020.1819608How to use a DOI?
Keywords
lattice Schwarzian Korteweg-de Vries equation, integrable symplectic map, finite genus solution
Abstract

Based on integrable Hamiltonian systems related to the derivative Schwarzian Korteweg-de Vries (SKdV) equation, a novel discrete Lax pair for the lattice SKdV (lSKdV) equation is given by two copies of a Darboux transformation which can be used to derive an integrable symplectic correspondence. Resorting to the discrete version of Liouville-Arnold theorem, finite genus solutions to the lSKdV equation are calculated through Riemann surface method.

Open Access

## 1. Introduction

Remarkable progress has been made in recent years in the study of discrete soliton equations (see  and the references therein). Among the related mathematical theories, the property of multi-dimensional consistency plays an important role in the understanding of discrete integrability. In the 2-dimensional case, it leads to the well-known Adler-Bobenko-Suris (ABS) list [1, 10], which gives a classification of integrable quadrilateral lattice equations. Quite a few works have appeared in the study of the ABS equations, concerning their relations with the usual soliton equations, the Lax pairs, explicit analytic solutions, Bäcklund transformations (BTs), symmetries and conservation laws etc. [3, 6, 7 , 13, 14, 19, 21, 22, 26, 32, 33 ].

The purpose of this paper is to investigate the lSKdV equation, which was first given in ,

Ξ:=γ12(uu^)(u˜u˜^)γ22(uu˜)(u^u˜^)=0,(1.1)
where the usual notation is adopted: u = u(m, n), u˜=u(m+1,n), û = u(m, n+1). Eq. (1.1) is exactly the Q1(0) model (Q1 with δ = 0) in the ABS hierarchy. The approach of Lax representation will be used to confirm the integrability of Eq. (1.1) and to calculate its basic explicit analytic solutions, the finite genus solutions [6, 7].

To produce a purely discrete Lax pair, it is vital to select two appropriate discrete spectral problems. It turns out that a special role is played by the semi-discrete integrable equations, which are also of independent interest, see  and references therein, where the well-known Toda, the Volterra and the Ablowitz-Ladik hierarchies are investigated thoroughly. A semi-discrete Lax pair can be constructed with the help of a continuous spectral problem and its Darboux transformation (DT), where the DT is regarded as a discrete spectral problem [4, 18], which usually leads to an integrable symplectic map by using the non-linearization technique . Refer to , integrable maps are called BTs whose geometrical explanation is given in terms of spectral curves and their Jacobians. And the symplectic correspondences (BTs) compatible with finite gap solutions of KdV have been discussed through DTs for the standard KdV spectral problem .

In our case we consider the continuous SKdV equation,

φyφx+14S[φ;x]=0,(1.2)
where S[ϕ;x] denotes the Schwarzian derivative of ϕ [20, 30, 31], i.e.
S[φ;x]=(φxxφx)x12(φxxφx)2,=φxxxφx32(φxxφx)2. (1.3)

Technically, it is more convenient to use the derivative version

wy+14(wxx3wx22w)x=0,(1.4)
with w = ϕx. The Eq. (1.4) has a Lax pair given by
xχ=(0wλ1w1λ10)χ,(1.5)
yχ=(wx2wλ2wλ3+14(wxx3wx22w)λ1w1λ3+14w2(wxxwx22w)λ1wx2wλ2)χ.(1.6)

We note that in , the Lax pair for one Schwarzian PDE, which is equivalent to the SKdV hierarchy via expansions on the independent variables and has a fully discrete counterpart (1.1) by considering the independent variables as lattice parameters, has been found. However, we have not been able to blend it with the algebro-geometric technique of nonlinearization employed in the present paper. Fortunately, here each of the linear systems (1.5) and (1.6) can be nonlinearized to produce an integrable Hamiltonian system. Thus we find a Liouville integrable system associated with a spectral problem (see Sec. 2) given by

xχ=U(λ,u)χ,U(λ,u)=(uλ0u),(1.7)
and find that the following DT of Eq. (1.7) is critical:
χ˜=(λ2γ2)1/2D(γ)(λ,b)χ,D(γ)(λ,b)=(λγbγb1λ).(1.8)

The compatibility condition Dx(γ)=U˜D(γ)D(γ)U gives rise to

bx/b=u+u˜,γb1=uu˜.(1.9)

This suggests a constraint b = γ/(uũ) and leads to a Lax pair, different from the one in , for Eq. (1.1).

### Lemma 1.1.

The lSKdV equation (1.1) has a Lax pair

χ˜=(λ2γ12)1/2D(γ1)(λ,b)χ,b=γ1/(uu˜),χ^=(λ2γ22)1/2D(γ2)(λ,b)χ,b=γ2/(uu^),(1.10)
with
D^(γ1)D^(γ2)D˜(γ2)D(γ1)=1ϒ((uu˜)(uu^)λ(u˜^u˜u^+u)0(u˜u˜^)(u^u˜^))Ξ,(1.11)
where ϒ=(uu˜)(uu^)(u˜u˜^)(u^u˜^) and Ξ is defined by Eq. (1.1).

The paper is organised as follows. In Sec. 2, a finite-dimensional Hamiltonian system which is a nonlinear version of the spectral problem (1.7) is presented. In Sec. 3, resorting to the Hamiltonian system, we construct an integrable symplectic map. In addition, with the help of the Burchnall-Chaundy theory, the discrete potential is expressed in terms of theta functions. In Sec. 4, based on the discrete version of the Liouville-Arnold theorem, the finite genus solutions of lSKdV equation (1.1) are obtained through the commutativity of integrable maps .

## 2. The Integrable Hamiltonian System (H1)

Take the symplectic manifold (ℝ2N, dp ∧ dq) as the phase space. The symplectic coordinate is defined as (p, q) = (p1,..., pN, q1,...,qN). Let A = diag(α1,...,αN) with distinct, non-zero α12,,αN2. Define a Lax matrix

L(λ;p,q)=σ++12j=1N(ɛjλαj+σ3ɛjσ3λ+αj)=(λQλ(p,q)1Qλ(Ap,p)Qλ(Aq,q)λQλ(p,q)),(2.1)
where σ+, σ3 are the usual Pauli matrices, and
ɛj=(pjqjpj2qj2pjqj),Qλ(ξ,η)=j=1Nξjηjλ2αj2,

∀ (ξ,η) = (ξ1,...,ξN1,...,η N) ∈ ℝ2N.

The generating function λ = detL(λ; p, q) is a rational function of the argument ζ = λ2 ,

λ(p,q)=(Qλ(Ap,p)1)Qλ(Aq,q)λ2Qλ2(p,q).(2.2)

The expansion λ=l=1Flζ1 gives rise to a set of quantities on phase space as

F1=Aq,qp,q2,Fl=A2l1q,q+j+k=lj,k1A2j1p,pA2k1q,qj+k=l+1j,k1A2j2p,qA2k2p,q,(2.3)
(l = 1, 2,...), where ξ,η=j=1Nξjηj. Consider the Hamiltonian system (H1), defined by the Hamiltonian function
H1=F12=12Aq,q12p,q2,x(pjqj)=(H1/qjH1/pj)=(p,qαj0p,q)(pjqj),1jN.(2.4)

They are exactly N copies of Eq. (1.7) with distinct λ = αj and the constraint

u=fU(p,q)=p,q.(2.5)

In this context (H1) is called a non-linearization of the linear spectral problem (1.7).

According to the Liouville-Arnold theory , we shall discuss the coefficients F1,...,FN given by (2.3) are first integrals of the phase flow with Hamiltonian function H1, i.e., {Fj, H1} = 0 (j = 1,...,N), where {·, ·} denotes the Poisson bracket on the phase space. The involution and functional independence between F1,...,FN guarantee that the Hamiltonian system (H1) is completely integrable.

Consider the Hamiltonian system (λ),

ddtλ(pjqj)=(λ/qjλ/pj)=W(λ,αj)(pjqj),W(λ,μ)=2λ2μ2(λL11(λ)μL12(λ)μL21(λ)λL11(λ))=L(λ)λμ+σ3L(λ)σ3λ+μ,(2.6)
where L(λ) is the abbreviation of L(λ; p, q) and Lij(λ), i, j = 1, 2 are entries of the matrix L(λ). Hence we obtain dεj/dtλ = [W(λ, αj), εj], where [·, ·] stands for the matrix commutator. Based on this formula, it is easy to derive the following basic equation,
ddtλL(μ)=[W(λ,μ),L(μ)],λ,μ.(2.7)

As a corollary, we have

{μ,λ}=0,λ,μ;(2.8)
{Fj,Fk}=0,j,k=1,2,.(2.9)

Actually, by Eq. (2.7), (d/dtλ)L2(μ) = [W(λ, μ), L2(μ)]. Since L2(μ) = −Iℱμ, where I is the identity matrix, we have dμ/dtλ = 0. According to the definition of Poisson bracket , this is exactly Eq. (2.8), whose power series expansion gives rise to Eq. (2.9).

The generating function λ has a factorization

λ=F1Z(ζ)α(ζ)=R(ζ)ζα2(ζ),(2.10)
with α(ζ)=j=1N(ζαj2), Z(ζ)=k=1N1(ζζk), R(ζ) = ζα(ζ)Z(ζ), where F1 is given by Eq. (2.3). The spectral curve is defined as
:ξ2R(ζ)=0,(2.11)
which is hyperelliptic with genus g = N − 1 and has two points at infinity, ∞+, ∞. At the branch point 𝔬 = (ζ = 0, ξ = 0), has a local coordinate λ = ζ1/2. The generic point on is given as
𝔭(ζ)=(ζ,ξ=R(ζ)),(τ𝔭)(ζ)=(ζ,ξ=R(ζ)),
where τ : is the hyperelliptic involution. The variables {νj2} defined as the roots of the equation
L21(λ)=j=1Nαjqj2λ2αj2=Aq,q𝔫(ζ)α(ζ)=0,𝔫(ζ)=j=1g(ζνj2),(2.12)
give an elliptic coordinate system . By Eq. (2.7) we have
ddtλL21(μ)=2(W21(λ,μ)L11(μ)W11(λ,u)L21(μ)).(2.13)

Putting μ = νk, with L11(νk)=F1R(νk2)/(νkα(νk2)) from Eq. (2.10), we get the evolution of the elliptic variables along the λ-flow,

12R(νk2)d(νk2)dtλ=2F1α(ζ)𝔫(ζ)(ζνk2)𝔫(νk2),1kg,(2.14)
k=1g(νk2)gs2R(νk2)d(νk2)dtλ=2F1α(ζ)ζgs,1sg,(2.15)
where the interpolation formula of polynomials is used. With the help of the quasi-Abel-Jacobi variables
φs=k=1g𝔭0𝔭(νk2)ωs,ωs=ζgsdζ2R(ζ),1sg,(2.16)

Eq. (2.15) is rewritten in a simple form and gives rise to

### Proposition 2.1.

The ℱλ- and the Fl-flow are linearized by ϕ′s as

dφsdtλ={φs,λ}=2F1α(ζ)ζgs,1sg,(2.17)
dφsdtl={φs,l}=2F1Als1,l=1,2,,(2.18)
where A0 = 1; Aj = 0 (j = 1, 2,...); while Aj (j = 1, 2,...), are defined by
ζNα(ζ)=1k=1N(1αk2ζ1)=j=0Ajζj.

In particular, {ϕ′s, F1} = 0, 1 ⩽ sg.

### Proposition 2.2.

The Hamiltonian system (H1) is integrable, possessing N integrals F1,...,FN, involutive with each other and functionally independent in the dense, open subset 𝒪 = {(p, q) ∈ ℝ2N : F1 ≠ 0}.

### Proof.

Fl is an integral since {H1, Fl} = (1/2){F1, Fl} = 0 by Eq. (2.9). It needs only to prove that dF1,...,dFN are linearly independent in T(p,q)*2N at (p, q) ∈ 𝒪. Suppose j=1NcjdFj=0. Then

c2{φs,F2}++cN{φs,FN}=0,1sN1.

By Eq. (2.18), the coefficient matrix is non-degenerate,

({φ1,F2}{φ1,FN}{φg,F2}{φg,FN})=2F1(1A1A2Ag11A1Ag21A11).

Thus c2 = ··· = cN = 0 and c1dF1 = 0. We have c1 = 0 since dF1 ≠ 0 at 𝒪. Otherwise,

12dF1=j=1N(p,qqjdpj+(αjqj+p,qpj)dqj)=0.

Hence αjqj + 〈p, qpj = 0, ∀j; and F1 = 0. This is a contradiction.

## 3. The Integrable Symplectic Map 𝒮γ

As a non-linearization of Eq. (1.8), define a map 𝒮γ : ℝ2N → ℝ2N, (p,q)(p˜,q˜) by

(p˜jq˜j)=(αj2γ2)1/2D(γ)(αj,b)(pjqj),1j,N,(3.1)
where a constraint b = fγ(p, q) is to be chosen so that 𝒮γ is integrable and symplectic.

### Lemma 3.1.

Let P(γ)(b; p, q) = b2L21(γ) + 2bL11(γ) − L12(γ). Then

L(λ;p˜,q˜)D(γ)(λ,b)D(γ)(λ,b)L(λ;p,q)=γb1P(γ)(b;p,q)σ3,(3.2)
j=1N(dp˜jdq˜jdpjdqj)=12γb2dP(γ)(b;p,q)db.(3.3)

### Proof.

By Eq. (3.1), we get ɛ˜jD(γ)(αj)D(γ)(αj)ɛj=0. Besides, we have σ32=I and

σ3D(γ)(λ)σ3=D(γ)(λ),D(γ)(±λ)D(γ)(αj)=±(λαj)I.

Based on these preparations, we calculate the left-hand side of Eq. (3.2),

[σ+D(γ)(λ)]+12j=1N(ɛ˜jD(γ)(λ)D(γ)(λ)ɛjλαj+σ3ɛ˜jσ3D(γ)(λ)D(γ)(λ)σ3ɛjσ3λ+αj)=γb1σ3+12j=1N((ɛ˜jɛj)+σ3(ɛ˜jɛj)σ3)=(γb1+p˜,q˜p,q)σ3.

By using Eq. (3.1), we obtain

γb1+p˜,q˜p,q=γb1P(γ)(b;p,q).(3.4)

This proves Eq. (3.2). Eq. (3.3) is obtained through direct calculations.

Consider the quadratic equation P(γ)(b) = 0, whose roots give the constraint on b,

b=fγ(p,q)1Qγ(Aq,q)(γQγ(p,q)±γ(p,q)).(3.5)

Actually γb can be written as a meromorphic function on ,

𝔟(𝔭)=1Qγ(Aq,q)(γ2Qγ(p,q)+F1ξα(γ)).

Though doubled-valued as a function of β ∈ ℂ, it is single-valued as a function of 𝔭(β2) ∈ . Hence we obtain

### Proposition 3.1.

The map 𝒮γ : ℝ2N → ℝ2N, (p,q)(p˜,q˜), defined as

(p˜jq˜j)=(αj2γ2)1/2(αjpj+γbqjγb1pj+αjqj)|b=fγ(p,q),1j,N,(3.6)
is symplectic and integrable, possessing the Liouville set of integrals
Fl(p˜,q˜)=Fl(p,q),1jN.(3.7)

### Proof.

Since P(γ)(b) = 0, by Eq. (3.2) and (3.3) we have

L(λ;p˜,q˜)D(γ)(λ,fγ(p,q))D(γ)(λ,fγ(p,q))L(λ;p,q)=0,(3.8)
j=1Ndp˜jdq˜j=j=1Ndpjdqj.(3.9)

Taking the determinant of Eq. (3.8), we obtain λ(p˜,q˜)=λ(p,q), hence Eq. (3.7).

By Eq. (3.7), the discrete flow (p(m),q(m))=𝒮γm(p0,q0) has constants of motion {Fl}. Define finite genus potentials as

b(m)=bm=fγ(p(m),q(m)),(3.10)
u(m)=um=fU(p(m),q(m))=p(m),q(m).(3.11)

By Eq. (3.4), they have the relation

bm=γ/(umum+1),(3.12)
which meets the requirement of Eq. (1.10). Along the m-flow, Eq. (3.8) is rewritten as
Lm+1(λ)Dm(γ)(λ)=Dm(γ)(λ)Lm(λ),(3.13)
where Lm(λ) = ; p(m), q(m), Dm(γ)(λ)=D(γ)(λ,bm). Now we calculate um with the help of the following spectral problem and its fundamental solution matrix Mγ(m, λ),
hγ(m+1,λ)=Dm(γ)(λ)hγ(m,λ);(3.14)
Mγ(m+1,λ)=Dm(γ)(λ)Mγ(m,λ),Mγ(0,λ)=I.(3.15)

By induction we have

Mγ(m,λ)=Dm1(γ)(λ)Dm2(γ)(λ)D0(γ)(λ),detMγ(m,λ)=(λ2γ2)m,Lm(λ)Mγ(m,λ)=Mγ(m,λ)L0(λ).(3.16)

### Lemma 3.2.

The following functions are polynomials of the argument ζ = λ2:

Mγ11(2k,λ),λ1Mγ12(2k,λ),λ1Mγ21(2k,λ)Mγ22(2k,λ),λ1Mγ11(2k+1,λ),Mγ12(2k+1,λ),Mγ21(2k+1,λ),λ1Mγ22(2k+1,λ).(3.17)

Besides, as λ → ∞,

Mγ(m,λ)=(λm[1+O(λ2)]O(λm1)O(λm1)λm[1+O(λ2)]).(3.18)

By Eq. (3.13), the solution space λ of Eq. (3.14) is invariant under the action of the linear operator Lm(λ), which has two eigenvalues ρλ±=±ρλ,

ρλ=λ=F1R(ζ)λα(ζ).(3.19)

They define a meromorphic function 𝔠(𝔭)=F1ξ(𝔭)/α(ζ(𝔭)) on with 𝔠(𝔭(λ2))=λρλ+, 𝔠((τ𝔭)(λ2))=λρλ. The corresponding eigenvectors satisfy

h±(m,λ)=(h±(1)(m,λ)h±(2)(m,λ))=Mγ(m,λ)(cλ(±)1),(3.20)
(Lm(λ)ρλ±)h±(m,λ)=0.(3.21)

Putting m = 0, we solve

cλ±=L011(λ)±ρλL021(λ)=L012(λ)L011(λ)ρλ,(3.22)
cλ+cλ=L012(λ)L021(λ),(3.23)
defining a meromorphic function 𝔠(𝔭) with 𝔠(𝔭(λ2))=λcλ+, 𝔠((τ𝔭)(λ2))=λcλ. As λ → ∞,
cλ±=p,q±F1Aq,q|p0,q0λ[1+O(λ2)].(3.24)

### Lemma 3.3 (Formula of Dubrovin-Novikov type).

(h+(1)h(1)h+(1)h(2)h+(2)h(1)h+(2)h(2))|(m,λ)=(λ2γ2)mL021(λ)(Lm12(λ)Lm11(λ)+ρλLm11(λ)ρλLm21(λ)),(3.25)
h+(2)(m,λ)h(2)(m,λ)=Aq,qmAq,q0(ζγ2)mj=1gζνj2(m)ζνj2(0).(3.26)

### Proof.

Using Eq. (3.16), we calculate the left-hand side of Eq. (3.25),

LHS=Mγ(m,γ)(cλ+cλcλ+cλ1)MγT(m,λ)=1L021(λ)Mγ(m,λ)[L0(λ)+ρλI](0110)MγT(m,λ)=1L021(λ)[Lm(λ)+ρλI]Mγ(m,λ)(0110)MγT(m,λ)=1L021(λ)[Lm(λ)+ρλI](0110)detMγ(m,λ)=RHS.

With the help of Eq. (2.12) and (3.25), Eq. (3.26) is verified by some calculations.

### Lemma 3.4.

As λ → ∞,

h±(1)(m,λ)=p,q0±F1Aq,q0λm+1[1+O(λ2)],(3.27)
h±(2)(m,λ)=p,qmF1p,q0F1λm[1+O(λ2)].(3.28)

### Proof.

Since h±(1)(m,λ)=Mγ11(m,λ)cλ±+Mγ12(m,λ), we have Eq. (3.27) in virtue of Eq. (3.18) and (3.24). By Eq. (3.25) we get

h+(1)h(2)|(m,λ)=(λ2γ2)mLm11(λ)+ρλL021(λ)=p,qm+F1Aq,q0λ2m+1[1+O(λ2)],h+(2)h(1)|(m,λ)=(λ2γ2)mLm11(λ)ρλL021(λ)=p,qmF1Aq,q0λ2m+1[1+O(λ2)].

Thus we obtain Eq. (3.28) by solving h±(2) and using Eq. (3.27).

From Eq. (3.20) we have

h±(2)(2k,λ)=(λcλ±)λ1Mγ21(2k,λ)+Mγ22(2k,λ),λh±(2)(2k+1,λ)=(λcλ±)Mγ21(2k+1,λ)+λMγ22(2k+1,λ).

By (Lemma 3.2) and the discussion on λcλ±, two meromorphic functions (the Baker functions) H(2)(2k, 𝔭) and H(2)(2k + 1, 𝔭) are defined on , respectively, with

H(2)(2k,𝔭(λ2))=h+(2)(2k,λ),H(2)(2k,(τ𝔭)(λ2))=h(2)(2k,λ),H(2)(2k+1,𝔭(λ2))=λh+(2)(2k+1,λ),H(2)(2k+1,(τ𝔭)(λ2))=λh(2)(2k+1,λ).(3.29)

### Proposition 3.2.

H(2)(2k, 𝔭) and H(2)(2k + 1, 𝔭) have the divisors respectively,

j=1g[𝔭(νj2(2k))𝔭(νj2(0))]+2k𝔭(γ2)k(++),j=1g[𝔭(νj2(2k+1))𝔭(νj2(0))]+(2k+1)𝔭(γ2)+𝔬(k+1)(++).(3.30)

### Proof.

From Eq. (3.26) and (3.29) we obtain

H(2)(2k,𝔭)H(2)(2k,τ𝔭)=Aq,q2kAq,q0(ζγ2)2kj=1gζνj2(2k)ζνj2(0),H(2)(2k+1,𝔭)H(2)(2k+1,τ𝔭)=Aq,q2k+1Aq,q0ζ(ζγ2)2k+1j=1gζνj2(2k+1)ζνj2(0),(3.31)
where 𝔭 = 𝔭(ζ). As 𝔭 → ∞±, by Eq. (3.28) and (3.29) we have
H(2)(2k,𝔭)=p,q2kF1p,q0F1ζk[1+O(ζ1)],H(2)(2k+1,𝔭)=p,q2k+1F1p,q0F1ζk+1[1+O(ζ1)].(3.32)

By these formulas it is easy to calculate the divisors.

By using the technique developed by Toda , based on the meromorphic differentials dlnH(2)(2k, 𝔭) and dlnH(2)(2k + 1, 𝔭), immediately we get

j=1g𝔭(νj2(0))𝔭(νj2(2k))ω+k(+𝔭(γ2)ω+𝔭(γ2)ω)0,(mod𝒯),j=1g𝔭(νj2(0))𝔭(νj2(2k+1))ω+(k+1)(+𝔭(γ2)ω+𝔭(γ2)ω)+𝔭(γ2)𝔬ω0,(mod𝒯),(3.33)
where ω=(ω1,,ωg)T are the normalized basis of holomorphic differentials on , while 𝒯 is the basic lattice spanned by the periodic vectors of [8, 11]. With the help of the Abel map 𝒜(𝔭)=𝔭0𝔭ω, the Abel-Jacobi variable is defined as
φ(m)=𝒜(j=1g𝔭(νj2(m))).(3.34)

This endows Eq. (3.33) with a clear geometric explanation.

### Proposition 3.3.

In the Jacobi variety J() = ℂg /𝒯, the discrete flow 𝒮γm is linearized by the Abel-Jacobi variable

φ(m)φ(0)+mΩγ+δmΩ0γ,(mod𝒯),(3.35)
where δ2k = 0, δ2k+1 = 1, and
Ωγ=12(𝔭(γ2)+ω+𝔭(γ2)ω),Ω0γ=Ωγ+0𝔭(γ2)ω.(3.36)

The meromorphic function H(2)(2k, 𝔭) is expressed by its divisor up to a constant factor

H(2)(2k,𝔭)=constθ[𝒜(𝔭)+φ(2k)+K]θ[𝒜(𝔭)+φ(0)+K]exp{k𝔭0𝔭ω[𝔭(γ2),+]+ω[𝔭(γ2),]},(3.37)
where K is the Riemann constant vector and w[𝔭, 𝔮] is an Abel differential of the third kind, possessing only two simple poles at 𝔭, 𝔮 with residues +1, −1, respectively. Resorting to Eq. (3.32), by the asymptotic behaviors of Eq. (3.37) near ∞± we obtain
u2kF1u0F1=constθ[𝒜(+)+φ(2k)+K]θ[𝒜(+)+φ(0)+K](rγ+rγ,+)k,u2k+F1u0+F1=constθ[𝒜()+φ(2k)+K]θ[𝒜()+φ(0)+K](rγrγ,+)k,(3.38)
where
rγ±=lim𝔭±1ζ(𝔭)exp𝔭0𝔭ω[𝔭(γ2),±],rγ,±=exp𝔭0±ω[𝔭(γ2),].(3.39)

We introduce a new variable vm by

vm=umF1um+F1,um=F11+vm1vm.(3.40)

Cancelling the constant factor in Eq. (3.38), we arrive at

v2k=v0θ[2kΩγ+Ω+K(0)]θ[K(0)]θ[2kΩγ+K(0)]θ[Ω+K(0)]e2kRγ,(3.41)
where Ω=+ω and
𝒜()=𝔭0ω=η,𝒜(+)=Ω+η,K(m)=φ(m)+K+η,Rγ=12ln(rγ+rγ,+rγrγ,+).(3.42)

Similarly, considering the analytic expression for H(2)(2k + 1, 𝔭) leads to

v2k+1=v0θ[(2k+1)Ωγ+Ω0γ+Ω+K(0)]θ[K(0)]θ[(2k+1)Ωγ+Ω0γ+K(0)]θ[Ω+K(0)]e(2k+1)Rγ+R0γ,(3.43)
where
R0γ=Rγ+lnr0γ,r0γ=exp+ω[𝔬,𝔭(γ2)].(3.44)

### Proposition 3.4.

The finite genus potential vm, defined by Eq. (3.11) and (3.40), has an explicit evolution formula along the discrete flow 𝒮γm,

vm=v0θ[mΩγ+δmΩ0γ+K(0)+Ω]θ[K(0)]θ[mΩγ+δmΩ0γ+K(0)]θ[K(0)+Ω]emRγ+δmR0γ,(3.45)
where the vectors K(m), Ωγ, Ω0γ and Ω are given by Eq. (3.36) and (3.42), while the constants Rγ, R0γ are defined by Eq. (3.42) and (3.44); moreover, δ2k = 0, δ2k+1 = 1, for all k.

## 4. Solutions of lSKdV equation (1.1)

Let γ1, γ2 be the two constants given in Eq. (1.1). By (Proposition 3.1), setting γ = γ1, γ2 in the above we have two symplectic maps 𝒮γ1 and 𝒮γ2, sharing the same set of integrals {Fl}. Resorting to the discrete version of Liouville-Arnold theorem [25,27,29], they commute. Thus we have well-defined functions with two discrete arguments m and n,

(p(m,n),q(m,n))=𝒮γ1m𝒮γ2n(p0,q0),bmn=fγ(p(m,n),q(m,n)),umn=fU(p(m,n),q(m,n))=p(m,n),q(m,n),vmn=(umnF1)/(umn+F1).(4.1)

### Proposition 4.1.

Both the functions umn and vmn, defined by Eq. (4.1), solve Eq. (1.1).

### Proof.

By the commutativity of 𝒮γ1m and 𝒮γ2n, we have

(p(m,n),q(m,n))=𝒮γ1m(p(0,n),q(0,n))=𝒮γ2n(p(m,0),q(m,0)).(4.2)

From Eq. (3.12) we obtain

bmn=γ1/(uu˜)=γ1/(uu^).(4.3)

By Eq. (3.6), χj = (pj(m, n), qj(m, n))T solves simultaneously

χ˜j=(αj2γ12)1/2D(γ1)(αj,bmn)χj,bmn=γ1/(uu˜),χ^j=(αj2γ22)1/2D(γ2)(αj,bmn)χj,bmn=γ2/(uu^).(4.4)

Thus umn satisfies Eq. (1.1) by Eq. (1.11). In order to prove that vmn is also a solution, it is sufficient to notice that (i) F1 is a constant of motion which is independent of m and n; (ii) Eq. (1.1) is invariant under the Möbius transformation uv given by Eq. (4.1).

Apply Eq. (3.45) to the flow 𝒮γ1m and 𝒮γ2n successively. By v00vm0vmn we obtain

### Proposition 4.2.

The lSKdV equation (1.1) has finite genus solutions

vmn=v00θ[mΩγ1+nΩγ2+δmΩ0γ1+δnΩ0γ2+K00+Ω]θ[K00]θ[mΩγ1+nΩγ2+δmΩ0γ1+δnΩ0γ2+K00]θ[K00+Ω]exp(mRγ1+nRγ2+δmR0γ1+δnR0γ2),(4.5)
and umn=F1(1+vmn)/(1vmn). Further, any Möbius transformation wmn = (a11vmn + a12)/(a21vmn + a22) solves Eq. (1.1), where ajk are constants.

## Acknowledgments

This work is supported by National Natural Science Foundation of China (Grant Nos. 11426206; 11501521), State Scholarship Found of China (CSC No. 201907045035), and Graduate Student Education Research Foundation of Zhengzhou University (Grant No. YJSXWKC201913). We would like to thank Prof. Frank W. Nijhoff and Prof. Da-jun Zhang for helpful discussions.